Best Harmonic and Superharmonic L1-Approximants in Strips Original Research Article

Let Ω denote the open strip (−1, 1)×Rn−1, where n⩾2. We completely solve the problem of characterizing a best harmonic L1-approximant to a subharmonic function s on Ω (all functions are assumed to be continuous and integrable on Ω). This characterization was previously known only under highly restrictive hypotheses on s. The approach of this paper is based, in part, on ideas used recently to solve the corresponding problem for the unit ball. However, the unboundedness of Ω presents difficulties which require the use of new techniques and recent results from other branches of harmonic approximation theory. Superharmonic L1-approximation of subharmonic functions is also treated.